A Criterion for the Normality of Polynomials over Finite Fields Based on Their Coefficients

An irreducible polynomial over $\Bbb F_q$ is said to be normal over $\Bbb F_q$ if its roots are linearly independent over $\Bbb F_q$. We show that there is a polynomial $h_n(X_1,\dots,X_n)\in\Bbb Z[X_1,\dots,X_n]$, independent of $q$, such that if an irreducible polynomial $f=X^n+a_1X^{n-1}+\cdots+a_n\in\Bbb F_q[X]$ is such that $h_n(a_1,\dots,a_n)\ne 0$, then $f$ is normal over $\Bbb F_q$. The polynomial $h_n(X_1,\dots,X_n)$ is computed explicitly for $n\le 5$ and partially for $n=6$. When $\text{char}\,\Bbb F_q=p$, we also show that there is a polynomial $h_{p,n}(X_1,\dots,X_n)\in\Bbb F_p[X_1,\dots,X_n]$, depending on $p$, which is simpler than $h_n$ but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group.